Shaffer et al.
Supplementary Methods
Artificial Egg Design - In addition to meeting the sensor requirements, loggers had to be small
and light enough to fit inside the eggs of the three species studied (Table S1 below). For scale,
Figure S1 shows a logger next to an artificial egg for each species. Due to size constraints, logger
orientation within an egg was different for auklets compared to gulls and albatrosses where the
x-axis of the logger was parallel to the long axis of auklet eggs, but for gull and albatross eggs,
the x-axis of the logger was perpendicular to the long axis of the egg. An adjustment for this
difference in axis orientation was made during data post-processing.
Artificial eggs were designed and created by students of Art and Industrial Design
Departments at San Jose State University, San Jose, California. Artificial eggs were made of
polystyrene plastic (3 mm thick; SABIC’s Innovative Plastics, Pittsfield, MA) vacuum formed to
the approximate size and shape of the natural eggs for each species (Figure S1). Artificial eggs
had a twisting lock mechanism to keep both halves together without using adhesive during
deployments. However, on 3 occasions, the locking mechanism failed and eggs came apart
while a bird was incubating the artificial egg. These logger data were either excluded entirely or
retained to the point where it was obvious from temperature recordings (prolonged and
sustained temperatures below 30°C), that an egg was no longer being tended. Artificial eggs
were filled with a combination of foam rubber and/or thermally neutral wire pulling lubricant
(Clear Glide, Ideal Industries Inc.) distributed evenly around the logger within the egg. This gave
the artificial eggs mass similar to that of real eggs. However, due to natural variability in egg
mass and size, some artificial eggs were 5-30% lighter than the original egg because all egg
loggers were activated and packaged in artificial eggs prior to removal of the original egg in a
nest, and it was not always possible to account for extreme variations in egg mass or size prior
to a deployment.
Table S1. Colony and nest characteristics for each bird species. Colony locations included
Southeast Farallon Island, CA for auklets, Año Nuevo Island, CA for gulls and Kaena Point, Oahu,
HI for albatrosses.
Species
Colony Location
Habitat Type
Nest Type
Clutch
N
Egg Mass
Egg Length
Egg Width
Size
(g)
(mm)
(mm)
Cassin’s auklets*
37.42°N, 122.68°W
Temperate
Burrow
1 egg
43
28.2 ± 241
45.6 ± 1.6
33.3 ± 1.1
Western gulls*
37.11°N, 122.33°W
Temperate
Surface
1-3 eggs
27
76 ± 3.4
70.05 ± 1.4
46.8 ± 0.5
Laysan albatrosses#
21.34°N, 158.16°W
Tropical
Surface
1 egg
10
270 ± 12
108.0 ± 4.3
67.0 ± 1.8
* - Order Charadriiformes, # - Procellariiformes, 41 - Astheimer, L.B (1986) Egg formation in Cassin’s auklet. Auk 103, 682–693 .
Determining Orientation from Raw Sensor Data - This section details the post-processing steps
that take the raw accelerometer and magnetometer measurements and convert them into
estimates of the egg orientation (expressed as 3-2-1 Euler angles [yaw, pitch, roll], or
quaternions, or a rotation matrix). Sensors were calibrated by removing any axis-misalignment,
scale factor, or bias effects on the accelerometer, and hard and soft iron effects on the
magnetometer. Calibrated sensor measurements were then used in an Extended Kalman Filter
(EKF) to estimate the egg orientation. Finally, orientation data were distilled into statistics
about the detected egg rotation events.
1
Shaffer et al.
Calibration - For accelerometer and magnetometer sensors, calibration reduced each
measurement to a unit vector pointing in the direction of gravity and the local magnetic field,
respectively. Due to axis mis-alignment, non-orthogonality, scale factor, and bias errors, as the
egg is moved through different orientations, the body x, y, and z axis measurements trace out
the surface of an ellipsoid. When properly calibrated the norm of each measurement lies on the
surfaces of a sphere of radius 1 (or in engineering units, a sphere of radius 1g = 9.8 m/s2, or
equal to the local magnetic field strength, which depends on latitude, longitude, altitude, and
time of year [42].
An iterative non-linear least squares fitting technique was used [43,44]. First, a subset of
the data was selected to use in the calibration. As biases on the accelerometers are typically
small, a threshold on the total acceleration magnitude of 0.025 g was used to filter for data that
was expected to be stationary. The data used for calibration was then uniformly selected from
the remaining subset of data that fell between 10% and 90% of the total time the logger was
recording in order to ignore disturbances related to the temperature and magnetic field as the
loggers were introduced or removed from the nest. Typically, 500 data points were used in the
calibration.
At each iteration, k, of the algorithm, a non-linear least squares solution was found for A
and B in equation 1.
(1)
where yi,k is the measurement vector at time i and iteration k,
and
with the a's and b's representing the nine calibration parameters being fit.
2
Shaffer et al.
An example set of magnetometer data before and after calibration is shown in Figure S5
below. Note the significant offset and distortion in the raw data that is corrected by the
algorithm to bring the data to the unit sphere. After the calibrations were performed, checks
were made to estimate quality of the calibration on the complete data set and to establish the
relative magnitude of the z-component of the magnetic field. First, the mean and standard
deviation of the norm of all the calibrated measurements were computed. This metric was
typically not as great for the accelerometer because it includes measurements at the time the
egg is moving and experiencing more than just the acceleration due to gravity. In most cases,
the mean of all the accelerometer readings was within 1% of 1g and the standard deviation was
less than 3%. For the magnetometer, the mean was typically within 0.5% of the calibrated field
strength and the standard deviation was less than 2% of the calibrated field strength. These
limits may be exceeded for short data sets where there was not sufficient coverage of the unit
sphere (a wide variety of orientations) represented in the data set.
Figure S5. Magnetometer data before (left) and after (right) calibration.
The second check computed the dot product between the calibrated magnetic field and
gravity vectors for the calibration data set. The vertical normalized magnetic field components
returned by the dot product were used in an EKF. No attempts were made to calibrate the
effect of temperature on the sensors. From recorded temperature data, we expected only small
variations in scale factors and offsets due to the relatively small (2-4°C, see Figures 5 & 6)
diurnal variations in temperature. Accuracy estimates indicated by the filter (described below)
were on the order of ±2° in roll and pitch and ±4° in yaw.
Initialization - The initial roll and pitch angles were estimated from the first accelerometer
measurements using equations 2 and 3. The initial yaw angle was estimated by minimizing the
sum squared error between the predicted and measured magnetic field strengths given the
initial roll and pitch angles.
3
Shaffer et al.
(2 & 3)
The initial Euler angles were then converted to the initial quaternion estimate with equation 4.
The covariance matrix was initialized to P0 = diag( 0.01 0.01 0.01 0.01)
(4)
We filtered out those instances where there was a high probability that the sensor was
measuring the movement of the egg in addition to gravity. The magnitude of the acceleration
was tested to be with 2.5% of 1g. It should be noted that the yaw angle is computed relative to
the local magnetic North, leading to the true field strength vector multiplying T in the lower half
of equation 5. A requirement of attitude quaternions is that they should always be unit norm
[44].
(5)
Estimating Rotation Statistics - After propagating the attitude state as a quaternion, the more
intuitive 3-2-1 Euler angles, often called yaw (Ψ), pitch (θ), and roll (ϕ) were computed using
equations 6-8. The Euler angle uncertainties were estimated from the quaternion covariance
matrix as PEuler = JEuler2quat Pk (JEuler2quat)T where JEuler2quat is the Jacobian of the Euler angles with
respect to the quaternions.
(6-8)
Data Analysis – All primary (above) and secondary post-processing of data was performed using
custom routines created in MATLAB (The Mathworks, Natick, MA USA). For comparison of
behaviors between different species, a number of statistics were computed following the post
processing. The main input required for these was to identify egg `turning events' and to
quantify how much the egg rotated during each event and how long it took. We applied Euler's
rotation theorem between the orientations at each successive time step to determine the axis
and minimum angle through which the egg rotated to get from one orientation to the next. This
4
Shaffer et al.
was done by computing the direction cosine matrices corresponding to the orientations at
times 1 and 2 (R1 and R2) using the transpose of equation 5 from above.
The incremental rotation matrix was calculated from these direction cosine matrices Rinc
= R1-1 R2 and the angle of rotation and body frame axis of rotation were calculated using
equations 9 and 10.
(9)
(10)
Due to sensor noise, the data were smoothed over a 1-minute window centered about the time
of interest (using a 61 point Hamming function). This window was long enough that noise
integrated to a small value, but actual rotations show up as significant angle differences. This
was in effect a heavily smoothed angular rate signal. We employed a threshold of 0.03 rad/s,
which was large enough to exceed the noise floor, but small enough to detect small rotation
changes. Rotation events were then identified as consecutive time windows where the
threshold was exceeded. For each of these identified events, Euler's theorem was reapplied to
find the minimum rotation angle between the initial and final events. For each rotation event,
we calculated the minimum rotation angle needed to go from the initial to final orientation,
and the start and end times of a rotation.
References
41.
Astheimer, LB (1986) Egg formation in Cassin’s auklet. Auk 103: 682–693.
42.
Maus S, Macmillan S, McLean S, Hamilton B, Thomson A, et al. (2010) The US/UK world
magnetic model for 2010-2015.
43.
Dorveaux E, Vissière D, Martin A-P, Petit N (2009) Iterative calibration method for
inertial and magnetic sensors. Decision and Control, 2009 held jointly with the 2009 28th
Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48 th IEEE Conference on
IEEE.
44
Crassidis JL, Markley FL, Cheng Y (2007) Survey of nonlinear attitude estimation
methods. J Guid Control Dynam 30: 12-28.
5